---
title: 'ANASTAARS: Noise-Aware Quantum Optimizer'
url: https://www.emergentmind.com/topics/anastaars
type: topic
---

# ANASTAARS: Noise-Aware Quantum Optimizer

Searching arXiv for the ANASTAARS paper and closely related acronyms to ground the article in current arXiv records.
ANASTAARS, expanded as **A Noise-Aware Stochastic Trust-region Algorithm using Adaptive Random Subspaces**, is a derivative-free classical optimizer for variational quantum algorithms, especially the quantum approximate optimization algorithm (QAOA), in regimes where the ansatz depth is large and the parameter dimension $d=2p$ becomes difficult to manage. It is designed for shot-noisy objective functions and addresses the simultaneous problems of scaling, stochastic measurement error, and the cost of repeated circuit evaluations by combining stochastic trust-region methodology, adaptive random subspace models, selective reuse of previously acquired measurements, and an explicitly noise-aware acceptance test [2507.10992].

## 1. Definition and optimization setting

ANASTAARS is formulated for the QAOA setting in which the $p$-layer state is
\[
\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,
\]
with parameter vectors
\[
\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.
\]
The associated classical optimization problem is written as minimizing the expected problem-Hamiltonian energy,
\[
\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),
\]
or, after defining
\[
x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,
\]
as the stochastic program
\[
\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].
\]
Here $f_\theta(x)$ is a noisy shot-based realization of the cost function [2507.10992].

The optimizer is motivated by a specific failure mode of high-depth QAOA workflows. As $p$ grows, the dimension $d=2p$ grows correspondingly; full-space model-based derivative-free optimization becomes expensive because model construction scales polynomially with $d$; gradient-based schemes are often shot-expensive because gradients must be estimated; and deterministic trust-region methods can stagnate in the presence of shot noise. ANASTAARS is intended to mitigate all of these effects in a single framework. This suggests a design objective that is not merely asymptotic dimensionality reduction, but practical shot-efficiency under NISQ-era measurement constraints [2507.10992].

## 2. Core algorithmic principle: adaptive random subspace trust regions

The defining mechanism of ANASTAARS is optimization in low-dimensional random affine subspaces rather than in the full parameter space. At iteration $k$, with incumbent $x_k\in\mathbb{R}^d$, the method chooses a matrix
\[
Q_k \in \mathbb{R}^{d\times q_k},
\]
whose columns span a $q_k$-dimensional random subspace. The corresponding affine search space is
\[
\mathcal{Y}_k = \{x_k + Q_k s : s\in\mathbb{R}^{q_k}\}.
\]
A local model $m_k(s)$ approximates the objective restricted to this affine space,
\[
m_k(s)\approx f(x_k + Q_k s),
\]
and, for quadratic models, takes the form
\[
m_k(s)=\mathfrak{f}_k+\hat g_k^\top s+\frac12 s^\top \hat H_k s.
\]
The trust region is defined in subspace coordinates by
\[
\mathcal{B}(x_k,\Delta_k;Q_k) = \{x_k+Q_k s\in \mathcal{Y}_k:\|s\|\le \Delta_k\},
\]
and the trial step satisfies
\[
s_k \approx \arg\min_{s\in\mathbb{R}^{q_k},\ \|s\|\le \Delta_k} m_k(s).
\]
This structure replaces a full-space local model with a model in dimension $q_k\ll d$ whenever possible [2507.10992].

The method’s distinctive novelty is not only the use of random subspaces, but the **adaptive** enlargement of those subspaces after unsuccessful iterations. If the previous iteration was successful, or if further enlargement would exceed $q_{\max}$, the algorithm resets to $q=q_0$, draws a fresh random subspace, and builds a new $q_0$-dimensional model. If the previous iteration was unsuccessful, it augments the previous subspace by one orthogonal direction instead of discarding it. If the previous basis is represented by $U_k\in\mathbb{R}^{d\times q}$, the new direction is sampled as
\[
\mu_{k+1}\in U_k^\perp \cap \mathbb{S}^{d-1},
\]
and the augmented subspace is defined by
\[
Q_{k+1} = \sqrt{\frac{d}{q+1}\,[U_k,\mu_{k+1}] \in \mathbb{R}^{d\times (q+1)}.
\]
Operationally, this means failure triggers a richer local model in a strictly larger subspace rather than repeated optimization within the same inadequate slice of parameter space [2507.10992].

## 3. Random embeddings, interpolation models, and sample reuse

The probabilistic justification for ANASTAARS rests on the requirement that the random subspace retain enough alignment with a useful descent direction. The desired condition is
\[
\|Q_k^\top \nabla f(x)\| \ge \varepsilon \|\nabla f(x)\|.
\]
The paper uses the concept of a $WAM(\varepsilon,\beta)$ random matrix sequence satisfying
\[
\mathbb{P}\!\left(\|Q_k^\top v\|\ge \varepsilon \|v\|\right)\ge 1-\beta \quad\text{for all }v\in\mathbb{R}^d.
\]
This is supported through Johnson–Lindenstrauss transforms. Specifically, for suitable $q$,
\[
\mathbb{P}\!\left( (1-\varepsilon)\|v\|^2 \le \|Q^\top v\|^2 \le (1+\varepsilon)\|v\|^2 \right)\ge 1-\beta,
\]
and a Haar-based construction is used: if $\underline U\in \mathbb{O}(d)$ is Haar distributed and $U_q$ is its first $q$ rows, then
\[
Q := \sqrt{\frac{d}{q}\, U_q^\top
\]
has the desired JL-style property when
\[
q \ge \frac{4}{\eta}\varepsilon^{-2}\log(2/\beta)
\]
for an absolute constant $\eta>0$ [2507.10992].

Within each subspace, ANASTAARS builds interpolation or surrogate models. The paper presents three model families.

For **linear subspace interpolation**, given a poised set
\[
S^q=\{s_k^0,s_k^1,\dots,s_k^q\}\subset \mathbb{R}^q\cap \mathcal{B}(0,\Delta_k), \qquad s_k^0=0,
\]
the shot-averaged estimates are
\[
f_k(s_k^i;\theta_k^i) = \frac{1}{\pi_k}\sum_{\ell=1}^{\pi_k} f_{\theta_{k,\ell}^i}(x_k+Q_k s_k^i),
\]
and the linear model is
\[
m_k(s)=a_k^0 + a_k^\top s.
\]
The coefficient vector is obtained through the interpolation system
\[
L_k^\top a_k = \delta^{f_k(S^q;\theta_k)},
\qquad
L_k=[s_k^1-s_k^0\ \cdots\ s_k^q-s_k^0]\in\mathbb{R}^{q\times q}.
\]
After a failed iteration, the old interpolation points are embedded in the expanded subspace so that old physical points in $\mathbb{R}^d$ are preserved:
\[
x_k + Q_k s = x_k + Q_{k+1} s'.
\]
The new set is constructed as
\[
S_{k+1}^{q+1} = \left\{ \begin{bmatrix} \hat q\, s\\ 0 \end{bmatrix} : s\in S_k^q \right\} \cup \left\{ s_{k+1}^{q+1} = \begin{bmatrix} 0_q\\ \zeta_{k+1} \end{bmatrix} \right\},
\qquad
\hat q := \sqrt{1+\frac1q}.
\]
The resulting closed form shows that only one new point is needed to move from dimension $q$ to $q+1$:
\[
a_{k+1}^q = \frac{a_k}{\hat q}, \qquad a_{k+1}^1 = \frac{\delta^{f_{k+1}^{q+1}}{\zeta_{k+1}}, \qquad a_{k+1}^0 = a_k^0.
\]
This measurement reuse is one of the method’s central practical contributions [2507.10992].

For **minimum-Frobenius-norm quadratic models**, which are the main nonlinear models used in experiments, the paper defines $\nu(q)=\frac12(q+1)(q+2)$ and the polynomial bases
\[
\bar\Phi_L^q(s) = (1,s_1,\dots,s_q),\qquad
\bar\Phi_Q^q(s)=\left(\frac12 s_1^2,s_1s_2,\dots,\frac12 s_q^2\right).
\]
The model is
\[
m_k(s)=\alpha_L^\top \bar\Phi_L^q(s)+\alpha_Q^\top \bar\Phi_Q^q(s),
\]
where $(\alpha_L,\alpha_Q)$ solve
\[
\min_{\alpha_L,\alpha_Q}\ \frac12 \|\alpha_Q\|^2
\]
subject to interpolation constraints. After a failed iteration and dimension increase, the $(q+1)$-dimensional MFN model is formed analogously, again adding only one new function value while reusing the previous interpolation data [2507.10992].

The paper also presents a **diagonal-Hessian quadratic model** with $2q+1$ parameters, based on the set
\[
S^q = \{0\}\cup\{e_q^i\}_{i=1}^q\cup\{-e_q^i\}_{i=1}^q
\]
and basis
\[
\Phi^q(s)=\left(1,s_1,\dots,s_q,\frac12 s_1^2,\dots,\frac12 s_q^2\right).
\]
This cheaper construction yields a quadratic model with diagonal Hessian [2507.10992].

## 4. Noise awareness and trust-region acceptance

ANASTAARS is explicitly noise-aware. The objective is available only through shot-based realizations $f_\theta(x)$, and the function estimates at the current and trial points are written as
\[
\bar f_k^0 = \frac{1}{\pi_k}\sum_{\ell=1}^{\pi_k} f_{\theta_\ell^{0}(x_k), \qquad
\bar f_k^s = \frac{1}{\pi_k}\sum_{\ell=1}^{\pi_k} f_{\theta_\ell^{s}(x_k+Q_k s_k).
\]
These are required to be $\varepsilon_F$-accurate in the stochastic trust-region sense,
\[
|\bar f_k^0 - f(x_k)| \le \varepsilon_F \Delta_k^2, \qquad
|\bar f_k^s - f(x_k+Q_k s_k)| \le \varepsilon_F \Delta_k^2,
\]
though the paper emphasizes that such bounds hold only probabilistically because $f$ is not directly observed [2507.10992].

The local noise magnitude is estimated from repeated samples at the current point using the sample standard deviation
\[
\sigma_k^2 = \frac{1}{\pi_k-1} \sum_{\ell=1}^{\pi_k} \left( f_{\theta_\ell^{0}(x_k)-\bar f_k^0 \right)^2.
\]
The algorithm explicitly allows reuse of samples from previous iterations when estimating this noise level. Rather than using the standard ratio
\[
\rho_k = \frac{\bar f_k^0-\bar f_k^s}{m_k(0)-m_k(s_k)},
\]
ANASTAARS employs the modified, noise-aware ratio
\[
\tilde\rho_k = \frac{\bar f_k^0-\bar f_k^s + r\,\sigma_k}{m_k(0)-m_k(s_k)}.
\]
The acceptance rule requires both
\[
\tilde\rho_k \ge \eta_1
\quad\text{and}\quad
\|s_k\| \ge \eta_2 \Delta_k.
\]
If the step is accepted,
\[
x_{k+1}=x_k+Q_k s_k, \qquad \Delta_{k+1}=\min\{\gamma \Delta_k,\bar\Delta\};
\]
otherwise,
\[
x_{k+1}=x_k, \qquad \Delta_{k+1}=\gamma^{-1}\Delta_k,
\]
and a failure flag is set so that the next iteration attempts a larger subspace [2507.10992].

This coupling of trust-region logic to an estimated noise magnitude is the method’s main robustness mechanism against shot noise. A plausible implication is that the method is designed to avoid two opposite pathologies: accepting illusory improvements caused by stochastic fluctuation, and rejecting genuine progress because a noisy estimate makes the apparent decrease too small.

## 5. Scalability, shot efficiency, and computational rationale

ANASTAARS is presented as a response to the cost of building full-space stochastic interpolation models in large parameter dimensions. The paper states that, in a simple noise framework, full-space stochastic model construction may require at least
\[
\Omega\!\left((d+1)\max\{\Delta^{-2},\Delta^{-4}(1-\alpha^{1/(d+1)})\}\right)
\]
function evaluations, whereas a $q$-dimensional subspace construction replaces $d$ by $q\ll d$ [2507.10992].

The practical shot-saving mechanisms are described explicitly. First, the algorithm works in dimension $q\ll d$. Second, the subspace model has far fewer parameters than a full-space model. Third, after a failed iteration it does not discard previous interpolation data. Fourth, in the linear and MFN constructions, increasing the model dimension from $q$ to $q+1$ can require only one new function evaluation. Since every function value is itself obtained through repeated circuit executions, these design choices directly target shot cost rather than only floating-point cost [2507.10992].

The paper situates ANASTAARS relative to three optimizer families. It lies between classical model-based derivative-free trust-region methods such as BOBYQA and NEWUOA, stochastic or noise-aware derivative-free methods such as ANATRA, and random subspace methods such as STARS. Its distinguishing combination is: stochastic trust-region methodology, adaptive random subspace models, sample reuse across iterations, and explicit noise awareness [2507.10992].

The algorithmic pseudocode reflects this hybrid position. The required parameters are
\[
\gamma>1,\quad \eta_1\in(0,1),\quad \eta_2,r>0,\quad \Delta_0\in(0,\bar\Delta),\quad q_{\max}\in\{2,\dots,d\},\quad q_0\in\{1,\dots,q_{\max}\},
\]
and the method cycles through subspace construction, approximate trust-region minimization, estimation of $\bar f_k^0$, $\bar f_k^s$, and $\sigma_k$, then success or failure updates. The paper does not state a separate formal stopping rule in the excerpted pseudocode, and it does not present a new standalone convergence theorem for ANASTAARS itself. Its theoretical support is instead inherited primarily from the STARS/random-subspace framework and the JL-style probabilistic embedding guarantees [2507.10992].

## 6. Numerical experiments, benchmark behavior, and limitations

The experimental variant studied is **ANASTAARS-QD2**, which uses MFN quadratic subspace models, adaptive dimensions $q\in\{q_0,\dots,q_{\max}\}$, $q_0=2$, and $q_{\max}=d$. The parameters used in the experiments are
\[
r=1,\quad \gamma=2,\quad \eta_1=0.01,\quad \eta_2=0.9,\quad \varepsilon_F=5,\quad \Delta_0=1.
\]
The test problems are MaxCut instances on a toy graph with MaxCut value $6$ and on the Chvátal graph with MaxCut value $20$ [2507.10992].

The initial experiments use $p=5$ QAOA layers, so $d=2p=10$, on the Qiskit QASM simulator with per-evaluation shot counts
\[
B=\pi_k\in\{50,100,500,1000\},
\]
and results are reported over 30 trials. Baselines include STARS-QD2, PyBOBYQA, ImFil, NOMAD, ANATRA, and NEWUOA; in the larger-scale experiments, the reported comparison is mainly against ANASTAARS-QD2, NEWUOA, and NOMAD [2507.10992].

For $d=10$ ($p=5$), the paper reports that ANASTAARS-QD2 matches the best long-term performance achieved by PyBOBYQA on both the toy and Chvátal graphs, and that, unlike STARS-QD2, it does not easily get stuck. The paper attributes this difference to adaptive subspace growth after unsuccessful iterations. At $B=1000$ and large shot budget, it is described as among the best optimizers in trial quantiles [2507.10992].

The scalability study extends to
\[
p=5,\dots,25, \qquad d=10,\dots,50.
\]
Across shot counts $50,100,500,1000$, the reported qualitative findings are that in the smallest-budget cases the optimizer’s lead increases as dimension increases, and in the largest-budget cases it maintains its lead as dimension grows. Performance is described as consistent across shot regimes, from low-shot to high-shot settings. The paper’s overall conclusion is that ANASTAARS-QD2 provides a path toward larger-scale QAOA optimization than is common in the cited literature [2507.10992].

The stated limitations are equally important. The experiments are performed on simulated MaxCut QAOA rather than on hardware. The tested implementation is specifically ANASTAARS-QD2, not every possible model variant. The method introduces several hyperparameters, including $(q_0,q_{\max},\gamma,\eta_1,\eta_2,r,\Delta_0)$. Subspace success remains probabilistic because it depends on alignment between the sampled subspace and useful descent directions. The paper also does not state a strong new convergence theorem for ANASTAARS in the provided text [2507.10992]. These caveats suggest that the method’s current contribution is primarily algorithmic and empirical rather than a completed theoretical characterization.

## 7. Name disambiguation and related acronyms

The acronym **ANASTAARS** should be distinguished from several similarly named but unrelated arXiv topics. **A-STAR**, expanded as **The All-Sky Transient Astrophysics Reporter**, is a proposed high-cadence, wide-field X-ray transient survey mission for multimessenger astronomy, with instruments named Owl and Lobster and a survey strategy covering the available sky twice per 24 hours; it is not a quantum optimization method [1302.2542]. **ANTARESS**, expanded as **Advanced and Neat Techniques for the Accurate Retrieval of Exoplanetary and Stellar Spectra**, is a workflow for high-resolution exoplanet transit and occultation spectroscopy, optimized for extracted 2D echelle spectra and spatially resolved stellar-spectrum extraction; it is likewise unrelated to QAOA optimization [2407.19012]. The Antarctic optical survey program **AST3-2** concerns a robotic time-domain telescope at Dome A, Antarctica, with public release of processed images, catalogs, and light curves from its 2016 survey; this is an astronomical survey infrastructure result rather than an optimizer [2302.06997].

This distinction matters because acronym similarity could suggest a common research lineage where none exists. Within the supplied arXiv record, ANASTAARS refers specifically to a noise-aware adaptive random-subspace trust-region optimizer for variational quantum algorithms, and particularly for QAOA in shot-noisy, moderate-to-large parameter regimes [2507.10992].

Source: https://www.emergentmind.com/topics/anastaars